Electrical transport measurements on ultrathin single-crystalline Au nanowires, synthesized via a wet chemical route, show an unexpected insulating behavior. The linear response electrical resistance exhibits a power-law dependence on temperature. In addition, the variation of current over a wide range of temperature and voltage obeys a universal scaling relation that provides compelling evidence for a non-Fermi liquid behavior. Our results demonstrate that the quantum ground state in ultrathin nanowires of simple metallic systems can be radically different from their bulk counterparts and can be described in terms of a Tomonaga-Luttinger liquid (TLL), in the presence of remarkably strong electron-electron interactions.